Tensor spherical

Nuclear spin relaxation is caused by fluctuating interactions involving nuclear spins. We write the corresponding Hamiltonians (which act as perturbations to the static or time-averaged Hamiltonian, detemiming the energy level structure) in tenns of a scalar contraction of spherical tensors ... 

The five second-moment spherical tensor components can also be calculated and are defined as the quadrupolar polarization terms. They can be seen as the ELF basin equivalents to the atomic quadrupole moments introduced by Popelier [32] in the case of an AIM analysis ... 

Table 1 NMR interactions in irreducible spherical tensor form ...

The Hamiltonian describing the quadrupolar interaction in the laboratory frame (L), in the units of radians/s, can be written using the spherical tensor formalism as [1,6, 24]... 

One defines2 a spherical tensor of rank k by the commutation relations [/z,rf] = K7f ... 

It has six independent components. It is convenient to separate the trace of this tensor from the rest and thus introduce spherical tensors... 

With these definitions the creation operators a, rcj) transform as spherical tensors under rotation. The annihilation operators do not. However, it is easy to construct operators that do transform as spherical tensors [Eq. (1.23)]. These will be denoted by a tilde and written as... 

The algebra of U(4) can be written in terms of spherical tensors as in Table 2.1. This is called the Racah form. The square brackets in the table denote tensor products, defined in Eq. (1.25). Note that each tensor operator of multipolarity X has 2X+ 1 components, and thus the total number of elements of the algebra is 16, as in the uncoupled form. 

As discussed in Ref. [1], we describe the rotation of the molecule by means of a molecule-fixed axis system xyz defined in terms of Eckart and Sayvetz conditions (see Ref. [1] and references therein). The orientation of the xyz axis system relative to the XYZ system is defined by the three standard Euler angles (6, (j), %) [1]. To simplify equation (4), we must first express the space-fixed dipole moment components (p,x> Mz) in this equation in terms of the components (p. py, p along the molecule-fixed axes. This transformation is most easily done by rewriting the dipole moment components in terms of so-called irreducible spherical tensor operators. In the notation in Ref. [3], the space-fixed irreducible tensor operators are... 

The summation index n has the same meaning as in Eq. (31), i.e., it enumerates the components of the interaction between the nuclear spin I and the remainder of the system (which thus contains both the electron spin and the thermal bath), expressed as spherical tensors. are components of the hyperfine Hamiltonian, in angular frequency units, expressed in the interaction representation (18,19), with the electron Zeeman and the ZFS in the zeroth order Hamiltonian Hq. The operator H (t) is evaluated as ... 

The matrix elements (8.35) in the uncoupled space-fixed basis can be most easily evaluated if all interaction operators are represented as uncoupled products of spherical tensors, with each tensor defined in the space-fixed coordinate system. Since the Hamiltonian is always a scalar operator, we can write any interaction in the Hamiltonian as a sum... 

The total angular momentum basis is thus computationally more efficient, even for collision problems in external fields. There is a price to pay for this. The expressions for the matrix elements of the collision Hamiltonian for open-shell molecules in external fields become quite cumbersome in the total angular momentum basis. Consider, for example, the operator giving the interaction of an open-shell molecule in a 51 electronic state with an external magnetic field. In the uncoupled basis (8.43), the matrix of this operator is diagonal with the matrix elements equal to Mg, where is the projection of S on the magnetic field axis. In order to evaluate the matrix elements of this operator in the coupled basis, we must represent the operator 5 by spherical tensor of rank 1 (Sj = fl theorem [5]... 

With this notation, the electric charge qo of a monopole equals Qoo-Cartesian dipole components px, py, pz, are related to the spherical tensor components as Ql0 = pz, Qi i = +(px ipy)/y/2, with i designating the imaginary unit. Similar relationships between Cartesian and spherical tensor components can be specified for the higher multipole moments (Gray and Gubbins 1984). 

The factor C k of B(R) is often called the /th component of the Racah spherical tensor of rank k the three tensor components of rank 1 may be considered unit basis vectors spanning the (spherical) space. 

Table 53 Relationships among components of the Cartesian and spherical tensors [60]... 

A spherical tensor is called irreducible spherical tensor if its (2m + 1) components transform under a rotation R of the coordinate system according to... 

Lanthanide complexes with axial symmetry (i.e., possessing at least a threefold axis, see sect. 2.4.2) are exclusively considered because the principal magnetic z axis coincides with the molecular symmetry axis (Forsberg et al., 1995) and the c 2 spherical tensor operators do not contribute to the crystal-field potentials (Gorller-Walrand and Binne-mans, 1996). The rhombic term of Bleaney s approach V6B Hi (eqs. (42), (46)) thus vanishes and the crystal-field independent methods (eqs. (51), (53)) can be used without complications. 

Here Bk s stand for the crystal field parameters (CFP), and Ck(m) are one-electron spherical tensor operators acting on the angular coordinates of the mth electron. Here and in what follows the Wyboume notation (Newman and Ng, 2000) is used. Other possible definitions of CFP and operators (e.g. Stevens conventions) and relations between them are dealt with in a series of papers by Rudowicz (1985, 2000,2004 and references therein). Usually, the Bq s are treated as empirical parameters to be determined from fitting of the calculated energy levels to the experimental ones. The number of non-zero CFP depends on the symmetry of the RE3+ environment and increases with lowering the symmetry (up to 27 for the monoclinic symmetry), the determination of which is non-trivial (Cowan, 1981). As a result, in the literature there quite different sets of CFP for the same ion in the same host can be found (Rudowicz and Qin, 2004). 

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